What are the common factors of 18xy and 32xy?

Dec 8, 2016

The common factors with integer coefficients are:

$\pm 1 , \pm 2 , \pm x , \pm 2 x , \pm y , \pm 2 y , \pm x y , \pm 2 x y$

Explanation:

The prime factorisations of $18$ and $32$ are:

$18 = 2 \cdot {3}^{2}$

$32 = {2}^{5}$

Hence their common positive integer factors are:

$1 , 2$

If you include negative integer factors then they have common factors:

$\pm 1 , \pm 2$

Both $18 x y$ and $32 x y$ are multiples of $x$ and $y$, so ignoring scalar multipliers, their common polynomial factors are:

$1 , x , y , x y$

Putting all of the possible combinations together, all of their polynomial factors with integer coefficients are:

$\pm 1 , \pm 2 , \pm x , \pm 2 x , \pm y , \pm 2 y , \pm x y , \pm 2 x y$

The greatest common factor of $18 x y$ and $32 x y$ is $2 x y$